Optimal. Leaf size=129 \[ -\frac{\tan ^{-1}\left (\frac{2 \sqrt [4]{2} \sqrt{3 x^2+2}+2\ 2^{3/4}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt{3 x^2+2}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt{3}} \]
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Rubi [A] time = 0.0599293, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{3 x^2+2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{3 x^2+2}}{\sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[1/((2 + 3*x^2)^(1/4)*(4 + 3*x^2)),x]
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Rubi in Sympy [A] time = 73.9714, size = 92, normalized size = 0.71 \[ \frac{\sqrt [4]{2} \sqrt{3} i \sqrt{- x^{2}} \Pi \left (- i; \operatorname{asin}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{3 x^{2} + 2}}{2} \right )}\middle | -1\right )}{6 x} - \frac{\sqrt [4]{2} \sqrt{3} i \sqrt{- x^{2}} \Pi \left (i; \operatorname{asin}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{3 x^{2} + 2}}{2} \right )}\middle | -1\right )}{6 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(3*x**2+2)**(1/4)/(3*x**2+4),x)
[Out]
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Mathematica [C] time = 0.173179, size = 135, normalized size = 1.05 \[ -\frac{4 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )}{\sqrt [4]{3 x^2+2} \left (3 x^2+4\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )\right )-4 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((2 + 3*x^2)^(1/4)*(4 + 3*x^2)),x]
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Maple [F] time = 0.036, size = 0, normalized size = 0. \[ \int{\frac{1}{3\,{x}^{2}+4}{\frac{1}{\sqrt [4]{3\,{x}^{2}+2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(3*x^2+2)^(1/4)/(3*x^2+4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} + 4\right )}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 + 4)*(3*x^2 + 2)^(1/4)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 + 4)*(3*x^2 + 2)^(1/4)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [4]{3 x^{2} + 2} \left (3 x^{2} + 4\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(3*x**2+2)**(1/4)/(3*x**2+4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} + 4\right )}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 + 4)*(3*x^2 + 2)^(1/4)),x, algorithm="giac")
[Out]